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Sign-changing solutions for elliptic problems with singular gradient terms and \(L^{1}(\Omega )\) data

  • Stefano Buccheri
Article
  • 34 Downloads

Abstract

In this paper we deal with singular boundary value problems of the type
$$\begin{aligned} \qquad {\left\{ \begin{array}{ll} \displaystyle -\,\mathrm{div}\left( a(x,u)\nabla u\right) +b(x) \frac{|\nabla u|^{2} }{|u|^{\theta }}\text{ sign }(u) = f(x), &{} \text{ in } \Omega \text{, } \\ &{}\qquad \qquad \qquad \quad (0.1)\\ \qquad \qquad \qquad \quad \qquad \qquad \qquad \quad \;\, u = 0, &{} \text{ on } \partial \Omega \text{, } \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a open bounded set of \(\mathbb {R}^N\) with \(N>2\), a(xt) is a Carathéodory function with polynomial growth with respect to t, b(x) is bounded and measurable, \(\theta \in (0,1)\) and f(x) belongs to \(L^{1}(\Omega )\). The main concern is to consider sign-changing solutions outside the energy space \(W_0^{1,2}(\Omega )\).

Keywords

Quasilinear elliptic equations Singular gradient term Changing sign data 

Mathematics Subject Classification

35J62 35J75 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaSapienza Università di RomaRomeItaly

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